Densification strain rate

The linear densification strain rate kp is defined as (-1/3 V) dVIdt, which is used to substitute JQC) in Eq. (5.210), there is... 

The sintering equation for lattice diffusion can be derived in a similar way. The linear densification strain rate can also be derived from Eq. (5.216) by replacing rPgb(5gb with 2XDy, where D is the lattice diffusion coefhcient [71]. Because there is 0 = a l(nX ), the linear densification strain rate is given by... 

For matter transport by diffusion, a general equation for the linear densification strain rate can be expressed as... 

The derivation of the sintering equation for lattice diffusion follows a procedure similar to that described above for grain boundary diffusion. The linear densification strain rate can also be obtained from Eq. (8.116) by replacing irDgbSgb with IXDi, where Di is the lattice diffusion coefficient (65). Remembering that 4> = a /( aX ), these substitutions give... 

Hot pressing equations can be derived from the creep equations with certain modifications. In practices, sintering data are usually present as density versus time to obtain densification rate. Because sintering compacts are porous before they are fuUy densified, the creep equations developed for dense solids should be modified to be suitable for hot pressing equations. First, the creep rate that is a linear strain rate should be related to the densification rate which is a volumetric strain rate. Second, the porosity should be compensated. In hot pressing, the mass M of the powder and the cross-sectional area A of the die, are constants, while the density... 

In the case of thin film sintering the lateral constraint imposed by a rigid substrate allows a shrinkage only in the direction (z-direction) perpendicular to the film. The imposed constraint induces an in-plane tensile stress and the stress accelerates the shrinkage in the z-direction. From the lateral constraint Ex = Sy = 0, cr = 0, and cTx = cry = cr, and the expressions of the uniaxial strain rates in Eq. (5.33), the densification rate in the z-direction is expressed as... 

Taking Ip as approximately equal to the grain size G, and writing the densification rate in the form of a volumetric strain rate, Eq. (8.47) becomes... 

In the hot pressing of powders, the data are normally acquired in the form of density versus time from which the densification rate can be determined. Furtha--more, considerable porosity is present over a large part of the process. The modification of the creep equations derived for dense solids therefore attempt to incorporate these two factors (1) relating the creep rate (a linear strain rate) to the densification rate (a volumetric strain rate) and (2) compensating for the presence of porosity. 

The densification rate of the matrix is affected by the hydrostatic component of the stress in the matrix. For a free matrix, the linear densification rate (or strain rate) can be written [see Fq. (8.116)]... 

The free strain rate (i.e., the linear densification rate) is given by... 

Using Eqs. (11.34) and (11.37), the ratio AGJ3K can be calculated in terms of the relative density of the matrix and this ratio can be substituted into Eqs. (11.28), (11.29), and (11.35) to calculate the stresses and densification rates for the composite sphere model. Alternatively, for the self-consistent model, Gc can be found from Eq. (11.36), and the same procedure repeated to determine the stresses and strain rates. Figure 11.20 shows the predicted values for tjt as a function of the relative density of the matrix for the composite sphere and self-consistent models. For v less than 20 vol%, the predictions for the two models are almost identical, but they deviate significantly for much higher values of V,-. When v is less than —10-15 vol%, the predicted values of tJtT are not... 

Figure 12.9 Change in the densification strain per unit change in temperature as a function of density, determined from the data of Fig. 12.8. The data for the different heating rates fall on the same master curve. (From Ref. 9.)...

At first inspection Eq. 1 seems well suited to densification by hot pressing, in which case the creep strain rate would be replaced by the rate of densification. However, it leaves something to be desired in that the effect of porosity on the rate of densification is ignored in addition, this equation would predict a finite shrinkage rate at the end point condition of zero porosity, where shrinkage must obviously cease. 

The mechanical behaviour of a glass becomes more and more time dependent as the temperature is increased and as the viscous behaviour becomes predominant over the densification process (Westbrook, 1960 Wiederhom and Hockey, 1980 Le Bourhis and Metayer, 2000 Le Bourhis and Rouxel, 2003 Wilantewicz and Varner, 2008). Under large strain rates, the supercooled liquid shows non-Newtonian behaviour with an apparent viscosity decreasing with the strain rate amplitude (Simmons et al, 1988 Appendix J). Although this behaviour makes the process operations more difficult, it offers advantages since the operations can be carried out under lower stresses (for the same flow, rheofluidification) once high strain rates are achieved. 

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